Math Bridge: BLE ranging and path loss

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Math BridgeUX DesignStruggle-friendly runway

Why can 4 dB move a BLE estimate by 2.3 metres?

Follow radio spreading into a calibrated, uncertain distance estimate.

UX Uma, the guideUX Uma guides
The one targetTurn RSSI uncertainty into distance uncertainty.
The chapter case2.4 GHz; n=2; −59 dBm reference; 5 m; 4 dB error.
What it buys youA location UI that reports evidence instead of false precision.

A field team faces an unresolved physical question: Why can 4 dB move a BLE estimate by 2.3 metres? They must answer it before changing rssi error on the real device. Predict the direction first.

See the relationship before changing it

The figure reads from left to right. The blue card is rssi error. The middle card applies this page's relationship. The green card is free-space loss. Walk the arrows once: set the input, apply the rule, then read the result with its unit.

The retained audit below checks several chapter fixtures. This added model holds every other chapter fixture fixed, so the numeric fixture does not switch without explanation.

RSSI error changes free-space loss An input card leads through the page relationship to the free-space loss result. SET INPUT ONE CONTROL APPLY RULE predict calculate check units READ RESULT
Walk the arrows. The location display should widen its zone or lower confidence when RSSI becomes unstable.

Derive the baseline in four named moves

  1. 1

    Name the input. The chapter baseline for rssi error is 4.

  2. 2

    Name the relationship. FSPL(1 m, 2.4 GHz) = 40.1 dB calibration gap = 59.0-40.1 = 18.9 dB relative error = ln(10)x4/(10x2) = 0.461 distance error = 0.461x5 = 2.30 m

  3. 3

    Substitute the chapter fixture. Set rssi error to 4. The page ledger gives free-space loss as 40.05 dB.

  4. 4

    Read the result. Keep dB beside the value. Use it only inside the technical boundary on this page.

Predict, then change rssi error

Try Predict the direction of free-space loss. Move one control, calculate, then check your prediction.

4
Chapter baseline
Free-space loss

Observe The location display should widen its zone or lower confidence when RSSI becomes unstable. Reset the control to 4 and compare free-space loss.

Explain Only rssi error moves here. The other chapter fixtures remain fixed.

Check yourself

What should you do before trusting a moved-control result?
Answer: Predict its direction, apply the shown relationship, keep the units, and reset to the worked baseline.
What does this small model leave out?
Answer: Only rssi error moves. Field effects named in the page's technical boundary stay fixed.

Honesty note. This teaching model changes only rssi error. It replays the chapter fixture; it is not a field prediction.

1. Start with spreading

Radio power spreads over a larger area as distance grows. In free space, doubling distance spreads the same power over four times the area. That inverse-square rule becomes the path exponent n=2 in the chapter's log-distance model.

UX Uma: RSSI is a noisy clue shaped by the room, body, enclosure, and antenna.

2. Name every algebra move

1

Convert spreading to dBFSPL=20log10(d)+20log10(f)−147.55.

2

Compare calibrationSubtract ideal 1 m power from A.

3

Convert dB errorRelative range error ≈ ln(10)ΔRSSI/(10n).

4

Apply the rangeMultiply the fraction by claimed distance.

3. Reproduce the chapter claim

FSPL(1 m, 2.4 GHz) = 40.1 dB
calibration gap = 59.0−40.1 = 18.9 dB
relative error = ln(10)×4/(10×2) = 0.461
distance error = 0.461×5 = 2.30 m

An 8 dB wall plus 12 dB reliability allowance gives 20.0 dB fade margin. Buying 6 dB with power alone multiplies radiated power by 3.98×.

4. Try the RSSI error

TryMove measurement error and watch a sharp 5 m claim lose its precision.

RSSI error
Free-space loss
Ideal received
Calibration gap
Relative error
5 m error
Fade margin
Power for +6 dB

ObserveIn this small-error approximation, doubling dB uncertainty doubles distance uncertainty.

ExplainThe location display should widen its zone or lower confidence when RSSI becomes unstable.

Technical boundaries.

The log-distance model compresses a complicated room into fitted constants.

Exponent
n changes with the environment
Calibration
A changes with device, body, orientation, and enclosure
Error
Multipath is not always small, independent, or symmetric

Calibrate and validate in the installed route, not only on a clear bench.

5. Design the honest action

Choose a zone width from the action's consequence. A room hint can tolerate metres; an automatic door or safety action needs stronger ranging, multiple sources, or a guarded fallback.

6. Record the location contract

Store band, A, n, calibration place, device orientation, uncertainty, freshness, fade margin, battery cost, fallback, and the field route used to validate the estimate.

7. Check yourself

Where does n=2 come from?
Answer: Free-space power falls with the inverse square of distance.
What does 4 dB mean at 5 m?
Answer: About 46.1% relative error, or 2.30 m in this approximation.
Is −59 dBm a universal 1 m truth?
Answer: No. It is a device-and-environment calibration constant.
Honesty boundary.

The 4 dB, n=2, and 5 m case comes from the chapter; radio and margin constants are explicitly typical assumptions.

4 dB at 5 m
Chapter error claim
2.4 GHz and −59 dBm
BLE teaching case
8 dB + 12 dB
Illustrative margin ledger

Correct, not complete: RSSI arithmetic does not qualify a location service or high-impact action.

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